Is there a certain etiquette or something in general I should know about Orthodox Monasteries before I visit?
Visiting a monastery is a different kind of experience from one country to the next. I mainly stay at monasteries in Romania, and in fact will depart for the weekend to one tomorrow. Here they'll welcome you and demand no money from you, but you'll be expected to do work, from chopping wood to sanding down boards for icons or peeling potatoes. (Bringing a few kilograms of potatoes and leaving behind an envelope with some amount of cash would however go very well).
In Finland, on the other hand, you don't have to do anything in particular, but you will be charged a fairly high fee (minimum 30€) for each day you are there, and you might find it hard to see actual monks. The situation is similar for Bulgarian monasteries and for the Monastery of St Katherine in Sinai.
One thing that I didn't expect the first time I visited a monastery was that one must stay silent at mealtimes. The abbot appoints one of the brothers to read from Scripture, and the focus of everyone while eating should be on that. In general however there is no rule of silence in Orthodox monasteries and if you are doing common tasks with others you're free to make conversation.